Because of their connection with secant lines, tangents, and instantaneous rates, limits of the form $\lim _ { h \rightarrow 0 } \frac { f ( x + h ) - f ( x ) } { h }$ occur frequently in calculus. Evaluate this limit for the given value of $x$ and function $f$.
-$$( x ) = \frac { x } { 5 } + 7 , x = 10$$
A) Does not exist
B) $\frac { 1 } { 5 }$
C) 2
D) 9

Question

Quizplus · Accepted Answer

Substitute $x=10$ and evaluate the limit:
$$\lim_{h\rightarrow 0} \frac{f(10+h) - f(10)}{h} = \lim_{h\rightarrow 0} \frac{\frac{10+h}{5}+7 - \frac{10}{5}-7}{h}$$
Simplify:
$$= \lim_{h\rightarrow 0} \frac{h/5}{h} = \lim_{h\rightarrow 0} \frac{1}{5} = \frac{1}{5}.$$
Therefore, the answer is B.

Because of Their Connection with Secant Lines, Tangents, and Instantaneous lim⁡h→0f(x+h)−f(x)h\lim _ { h \rightarrow 0 } \frac { f ( x + h ) - f ( x ) } { h }limh→0​hf(x+h)−f(x)​

Because of Their Connection with Secant Lines, Tangents, and Instantaneous $\lim _ { h \rightarrow 0 } \frac { f ( x + h ) - f ( x ) } { h }$